1. 7.1 Geometric Mean 7.2 Pythagorean Theorem 7.3 Special Right Triangles 7.4 Trigonometry 7.5 Angles of Elevation & Depression 7.6 Law of Sines 7.7 Law of Cosines

2. 7.1 Geometric Mean Objective: We will be able to define the geometric mean between two numbers, and solve problems involving the relationships between parts of a right triangle and its altitude to its hypotenuse. PS is an _________. There is an interesting relationship ……. right altitude similar We say: “x“ is the _________________ between “k” and “m”. geometric mean The positive square root of the product of two numbers. example: Find x:

3. example: Find the geometric mean between - 1. 6 and 15 3. 25 and 7 We do not have the lengths of two sides, so we can not use the Pythagorean Theorem at first, but we can use geometric mean to find unknowns. example: find x, y, and z

4. 7.2 Pythagorean Theorem Objective: We will be able to use the Pythagorean Theorem & its converse to solve problems. How do we find out? Which side do we choose as the hypotenuse? A ______________ _________ is a set of 3 _____________ numbers that satisfy the _______________________ Theorem. *Use _____________ where ____ is the ______________ number. 1. 8, 16, 152. 52, 20, 48 Can these measures form the sides of a right triangle, and if so, can they be considered a Pythagorean Triple? Pythagorean Triple whole a 2 + b 2 = c 2 c largest

5. 7.3 Special Triangles Objective: We will use the properties of 45-45-90 and 30-60-90 triangles to solve problems. Start with a square… How do we find the length of the hypotenuse? _________________ Pythagorean Thm. a 2 + b 2 = c 2 n n n n h n 6 n 5 45-45-90 examples: find the remaining sides of the 45-45-90 triangles. ______ 15 x find x & y:

6. Start with an equilateral triangle… a 2 + b 2 = c 2 n 12 y 30-60-90 examples: find x and y in the 30-60-90 triangles. ______ y h n full side 2n is the hypotenuse REMEMBER! x y x 15 x 26

7. 7.4 Trigonometry Objective: We will be able to set up trigonometric ratios using right triangles, and be able to solve problems (find side lengths and angles) using trig. ratios. trigonometric When we compare the lengths of the sides of a right triangle, we use trigonometric ratios: _________, _________, and __________. cosine sinetangent These ratios can give us the measures of the angles! They are abbreviated as shown: _______ ( ) ___ ( ) ___________ ( ) ____________ ___________ = = = ____ Remember! “ ______ - ______ - ______ “ Greek letter for angle measure “theta” sin A = cos A = tan A = sin B = cos B = tan B = A C B o h a hypotenuse adjacent leg opposite leg hypotenuse adjacent leg opposite leg h h o o a a soh cah toa a = 4 b = 3 c = 5 A = B =

8. examples: Find the angleWhat if you know the angle already ? Keep the decimal ratio in your calculator during your calculations, but round your FINAL answer to the tenths!

9. 7.5 Angles of Elevation and Depression Objective: We will be able to trigonometry to solve problems involving angles of elevation and depression: real world applications! Angle of Elevation: The angle between the line of sight and the horizontal when an observer looks upward. Trigonometry in real world situations…….. Read the problem. Draw the picture Set up the trigonometric ratio Solve. Answer the question in a complete sentence. Angle of Depression: The angle measure from the horizontal line of sight down to the object. (Use alternate interior angle thm.) 125’ 70’ 35’ 200’